Answer :

Basic Idea:


A real function f is said to be continuous at x = c, where c is any point in the domain of f if :


where h is a very small ‘+ve’ no.


i.e. left hand limit as x c (LHL) = right hand limit as x c (RHL) = value of function at x = c.


This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :



NOTE: If f and g are two functions whose domains are same and both f and g are everywhere continuous then f/g is also everywhere continuous for all R except point at which g(x) = 0


As, f(x) =


Domain of f = { all Real numbers except 2 } = R – {–2}


Clearly it is not defined at x = –2, for rest of values it is continuous everywhere


Because 1 is everywhere continuous and x + 2 is also everywhere continuous


f(x) is everywhere continuous except at x = –2


f(f(x)) = f


Domain of f(f(x)) = R – {–2 , (}


For rest of values it just a fraction of two everywhere continuous function


at all other points it is everywhere continuous.


Hence,


f(f(x)) is discontinuous at x = –2 and x = –5/2


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